Alcohol abuse disorder or alcoholism is a common disorder that leads to brain defects and associated cognitive, emotional and behavioural impairments. Finding and extracting discriminative biological markers, which are correlated to healthy brain pattern and alcoholic brain pattern, helps us to utilize automatic methods for detecting and classifying alcoholism. Many brain disorders could be detected by analyzing the Electroencephalography (EEG) signals. In this paper, for extracting the required markers we analyze the EEG signals for two groups of alcoholic and control subjects. Then by applying wavelet transform, band-limited EEG signals are decomposed into five frequency sub-bands. Also, the principle component analysis (PCA) is employed to choose the most information carrying channels. By examining various features from different frequency sub-bands, six discriminative features for classification are selected. From functional brain network perspective, the lower synchronization in Beta frequency sub-band and loss of lateralization in Alpha frequency sub-band in alcoholic subjects are observed. Also from signal processing perspective we found that alcoholic subjects have lower values of fractal dimension, energy and entropy compared to control ones. Five different classifiers are used to classify these groups of alcoholic and control subjects that show very high accuracies (more than 90%). However, by comparing the performance of different classifiers, SVM, random forest and gradient boosting show the best performances with accuracies near 100%. Our study shows that energy and entropy of channel $C1$ in Alpha frequency sub-band are more important features for classification.